SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

On discreteness of spectrum and positivity of the Green’s function for a second order functional-differential operator on semiaxis

Sergey Labovskiy1* and Mário Frengue Getimane2

Author Affiliations

1 Department of Mathematics, Moscow State University of Economics, Statistics and Informatics, Nezhinskaya 7, Moscow, 119501, Russia

2 Instituto Superior de Transportes e Comunicações, Prolong. da Av. Kim Il Sung (IFT/TDM) - Edifício D1, Maputo, 2088, Mozambique

For all author emails, please log on.

Boundary Value Problems 2014, 2014:102  doi:10.1186/1687-2770-2014-102

Published: 7 May 2014

Abstract

We study conditions of discreteness of spectrum of the operator defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M2">View MathML</a>. The operator has two singularities at the ends of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M3">View MathML</a>. The second question is positivity of solutions of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M4">View MathML</a> under boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/102/mathml/M6">View MathML</a>. The used abstract scheme is close to the well-known MS Birman’s method in the spectral theory of self-adjoint operators. Conditions for discreteness of spectrum and positivity of the Green’s operator are obtained. The result relates to the MS Birman’s result on the necessary and sufficient condition for discreteness of spectrum of a polar-differential operation. The results may be interesting for researchers in qualitative theory of functional-differential equations and spectral theory of self-adjoint operators.

MSC: 34K08, 34K10, 34K12.

Keywords:
discreteness of spectrum; positive solutions; singular operator